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Even if Lagrange multiplier is an important mathematical tool, I haven't been used it up to now. It is simple than its look. When we need to optimize two variables with conditioning on the combination of the two variables.

An Example of Lagrange Multiplier

I describe the contents of video to help readers' understanding.

We want to minimize $C(x,y) = 6x^2 + 12y^2$ subject to $x+y=90$. By Lagrange multiplier, we can write the Lagrange multiplier equation as

$6x^2 + 12y^2 - \lambda(x+y-90)$.

Use the property of Lagrange multiplier that the partial derivatives of the above Lagrange multiplier equation should be equal to zero

$F_x = 12x - \lambda = 0$
$F_y = 24y - \lambda = 0$
$F_\lambda = x + y - 90 = 0$

From the first and the second equations, we will have $x = \lambda/12$ and $y = \lambda/24$. Putting these results to the third equation, we find

$\lambda/12 + \lambda/24 - 90=0$
$\lambda/24 = 30$
$\lambda = 720$,

which result that $x = 30*24/12 = 60$ and $y = 30*12/12 = 30$. Thus, the minimum of $C(x,y)$ becomes $6(60)^2 + 12(30)^2 = 32400$.

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